Advancements in sequencing have led to the proliferation of multi-omic profiles of human cells under different conditions and perturbations. In addition, many databases have amassed information about pathways and gene "signatures"-patterns of gene expression associated with specific cellular and phenotypic contexts. An important current challenge in systems biology is to leverage such knowledge about gene coordination to maximize the predictive power and generalization of models applied to high-throughput datasets. However, few such integrative approaches exist that also provide interpretable results quantifying the importance of individual genes and pathways to model accuracy. We introduce AKLIMATE, a first kernel-based stacked learner that seamlessly incorporates multi-omics feature data with prior information in the form of pathways for either regression or classification tasks. AKLIMATE uses a novel multiple-kernel learning framework where individual kernels capture the prediction propensities recorded in random forests, each built from a specific pathway gene set that integrates all omics data for its member genes. AKLIMATE has comparable or improved performance relative to state-of-the-art methods on diverse phenotype learning tasks, including predicting microsatellite instability in endometrial and colorectal cancer, survival in breast cancer, and cell line response to gene knockdowns. We show how AKLIMATE is able to connect feature data across data platforms through their common pathways to identify examples of several known and novel contributors of cancer and synthetic lethality.